We consider the existence of the periodic solutions in the
neighbourhood of equilibria for

We first give some definitions for our problem. A

This result can be used easily to Hamiltonian systems and obtain existence of periodic solutions. Later, many mathematicians were dedicated to study periodic solutions of Hamiltonian systems and tried to generalize the result. Gordon [

Bifurcation theory describes how the dynamics of systems change as parameters varied. The study of the Hamiltonian Hopf bifurcation has a long history. The Hamiltonian-Hopf bifurcation involves the loss of linear stability of a fixed point by the collision of two pairs of imaginary eigenvalues of the linearized flow and their subsequent departure off the imaginary axis. van der Meer [

Moreover, reversible systems have been studied for many years. Devaney [

In [

Motivated by [

Consider an equilibrium 0 of a

Here, the existence and the symmetric property of periodic solutions near the equilibrium point are all considered. The main idea is similar to [

We now consider the persistent occurrence of purely imaginary eigenvalues in equivariant Hamiltonian vector fields.

Let

Since we are interested in (partially) elliptic equilibria, we assume that

Since

Hoveijn et al. [

Consider a linear

If

If

Let

If

Now, we give the examples for the system (

When

In this paper, when

In this section, we introduce the main technique which is a Liapunov-Schmidt reduction. The Liapunov-Schmidt reduction here is similar to the one in [

Assume that a

Now, define an action

By the

Assume that

By (

Below, we will obtain that

By

Next, define a

Moreover, we can prove that

Then, solutions of the equation

Now, we will obtain that

If

Since

Lemma

Since the vector field

If the actions of

By Theorem 6.2 in [

In this paper,

In this section, we prove Theorem

Below, we consider the case that the equivariant symmetry

Note that dim ker

Since

By

Then, by the

We have that

Then, the symplectic form

Using the similar calculation and by (

Using (

We first give Lemma

Since

In the neighbourhood of the equilibrium

Since

Apparently, the symmetric periodic solutions should be twoparameters. But since the vector field

Next, we study the nonsymmetric periodic solutions in the neighbourhood of the equilibrium 0.

Except the symmetric Liapunov Center family described in Lemma

The proof can be divided into three cases:

By (

Now, we consider the real parts of (

We now consider the case that

The authors would like to thank the referees for their valuable comments and suggestions.